scholarly journals TTF-classes over perfect rings

1970 ◽  
Vol 11 (4) ◽  
pp. 499-503 ◽  
Author(s):  
J. S. Alin ◽  
E. P. Armendariz
Keyword(s):  
Torsion Class ◽  
Direct Sums

For a ring R with unit, let RM denote the category of unitary left R-modules. Following S.E. Dickson [3], a(non-empty) class of R-modules is a torsion class in rM if is closed under factors, extensions, and direct sums. If, in addition, is closed under submodules, then is said to be hereditary.

Q-Divisible Modules

1971 ◽  
Vol 14 (4) ◽  
pp. 491-494 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Quotient Ring ◽  
Torsion Class ◽  
Direct Sums ◽  
Factor Module ◽  
Left Quotient ◽  
Divisible Modules

Let R be a ring with 1 and let Q denote the maximal left quotient ring of R [6]. In a recent paper [12], Wei called a (left). R-module M divisible in case HomR (Q, N)≠0 for each nonzero factor module N of M. Modifying the terminology slightly we call such an R-module a Q-divisible R-module. As shown in [12], the class D of all Q-divisible modules is closed under factor modules, extensions, and direct sums and thus is a torsion class in the sense of Dickson [5].

Torsion Theories Induced by Tilting Modules

1984 ◽  
Vol 36 (5) ◽  
pp. 899-913 ◽  
Author(s):  
Ibrahim Assem
Keyword(s):  
Torsion Theory ◽  
Torsion Class ◽  
Tilting Module ◽  
Tilting Modules ◽  
Direct Sums ◽  
Commutative Field ◽  
Finite Dimensional ◽  
Torsion Theories ◽  
Image Position ◽  
Torsion Free

Let k be a commutative field, and A a finite-dimensional k-algebra. By a module will always be meant a finitely generated right module. Following [8], we shall call a module TA a tilting module if (1) pdTA ≦ 1, (2) Ext1A(T, T) = 0 and (3) there is a short exact sequencewith T’ and T” direct sums of direct summands of T. Given a tilting module TA, the full subcategories andof the category modA of A -modules are respectively the torsion-free class and the torsion class of a torsion theory on modA[8]. The aim of the present paper is to find conditions on a torsion theory in order that it be induced by a tilting module.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

PERFECTLY BOUNDED CLASSES OF ABELIAN GROUPS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250208 ◽  
Author(s):  
PATRICK W. KEEF
Keyword(s):  
Abelian Groups ◽  
Direct Sums ◽  
Proper Subclass ◽  
Infinite Height ◽  
Reduced Group

Let [Formula: see text] be the class of abelian p-groups. A non-empty proper subclass [Formula: see text] is bounded if it is closed under subgroups, additively bounded if it is also closed under direct sums and perfectly bounded if it is additively bounded and closed under filtrations. If [Formula: see text], we call the partition of [Formula: see text] given by [Formula: see text] a B/U-pair. We state most of our results not in terms of bounded classes, but rather the corresponding B/U-pairs. Any additively bounded class contains a unique maximal perfectly bounded subclass. The idea of the length of a reduced group is generalized to the notion of the length of an additively bounded class. If λ is an ordinal or the symbol ∞, then there is a natural largest and smallest additively bounded class of length λ, as well as a largest and smallest perfectly bounded class of length λ. If λ ≤ ω, then there is a unique perfectly bounded class of length λ, namely the pλ-bounded groups that are direct sums of cyclics; however, this fails when λ > ω. This parallels results of Dugas, Fay and Shelah (1987) and Keef (1995) on the behavior of classes of abelian p-groups with elements of infinite height. It also simplifies, clarifies and generalizes a result of Cutler, Mader and Megibben (1989) which states that the pω + 1-projectives cannot be characterized using filtrations.

scholarly journals The Beckenbach-Dresher inequality in the Ψ-direct sums of spaces and related results

2012 ◽  
Vol 2012 (1) ◽  
pp. 7 ◽  
Author(s):  
Ludmila Nikolova ◽  
Lars-Erik Persson ◽  
Sanja Varošanec
Keyword(s):  
Direct Sums

Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

2021 ◽  
Vol 56 (2) ◽  
pp. 343-374
Author(s):  
Boris Guljaš ◽  
Keyword(s):  
Hilbert Space ◽  
Hilbert Modules ◽  
Strict Topology ◽  
Direct Sums ◽  
Essential Ideal ◽  
Closed Submodule ◽  
Closed Submodules ◽  
Hereditary Algebras ◽  
Multiplier Algebras

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

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